外区域上的边值问题有着重要的应用，例如描述流体通过障碍物的运动、容量问题和声波或电磁波的散射模型等．本论文研究了两类非线性二阶椭圆方程的外边值问题的解的存在唯一性．

一类是由欧氏空间上的非欧范数所诱导的 Finsler N-Laplace 方程，我们研究了其在凸锥上的一个混合外边值问题．该问题来源于对各向异性介质里的对数容量的研究，被应用在拟共形几何、凝聚态物理和高能物理等领域．在无穷远处预定渐近对数行为的条件下，我们不仅运用变分方法证明了该问题的弱解的存在唯一性，而且进一步在 Serrin 型超定条件下，利用 Pohozaev 型等式和各向异性等周不等式的最佳刻画证明了弱解存在当且仅当问题的定义域为 Wulff 球的余集和凸锥的相交．这部分的研究结论已发表在 [CL2022]，是文献中首次对凸锥上的超定外问题的讨论．

另一类是 Hessian 型完全非线性椭圆方程的 Dirichlet 外问题．这类方程的典型例子包括了著名的 Monge--Amp`ere 方程，其在仿射几何、凸几何和最优传输等理论里有着重要的应用．我们首先考虑了方程右端项为常数的情形；在无穷远处预定渐近二次行为的条件下，我们利用比较原理和 Perron 方法证明了该问题粘性解的存在唯一性．这部分的主要结果已经发表在 [86]，是前人对 Hessian 方程、Hessian 商方程和特殊 Lagrangian 方程的相关研究的一般性推广．然后我们考虑了带有非常数右端项的 Hessian 商方程，对其在无穷远处预定渐近二次行为的 Dirichlet 外问题的可解性作了进一步讨论．

Boundary value problems in exterior domains occur in important applications,

for instance in the motion of fluids past an obstacle, capacity problems or scattering of acoustic or electromagnetic waves. This thesis is concerned with the existence and uniqueness of solutions to the exterior boundary value problem for two classes of nonlinear, second-order elliptic equations.

One is a mixed boundary value problem in convex cones for Finsler $N$-Laplace equations induced by non-Euclidean norms. It is motivated by the study of anisotropic logarithmic capacity which appears in quasiconformal geometry, condensed-matter and high-energy physics. Under a prescribed logarithmic condition at infinity, we not only establish the existence and uniqueness of weak solutions to this problem by applying variational method, but also prove that the problem admits a weak solution satisfying a Serrin-type overdetermined condition if and only if the domain where the problem is defined must be the intersection of the cone and the complement of a Wulff shape. Our approach is based on a Pohozaev-type identity and the characterization of

minimizers of the anisotropic isoperimetric inequality inside convex cones. These results were obtained in our paper [CL2022], originating the discussion in the literature for exterior overdetermined problems in convex cones.

The other is the exterior Dirichlet problem for Hessian type fully nonlinear elliptic equations, typical examples of which include the famous Monge--Amp\`ere equation that has found important applications in affine geometry, convex geometry and optimal transportation. We first consider the equations with a constant right hand side. Under a prescribed quadratic condition at infinity, we establish the existence and uniqueness theorem for viscosity solutions by exploiting comparison principles and Perron's method. This part is a reorganization of the main result in [JLL2021], extending previously known results for Hessian equations, Hessian quotient equations and special Lagrangian equations. Then we treat Hessian quotient equations with a non-constant right hand side. We also prove the solvability of its exterior Dirichlet problem with prescribed asymptotically quadratic behavior.